Skov came up through the youth academy at Silkeborg, making his league debut for the club on 16 May 2013 in a 1–1 home draw with Midtjylland.

Robert Skov played a key role in helping Silkeborg win the Danish 1st Division during the 2013/14 season, despite being only 17 years old.

[7] During the 2016–17 Danish Superliga season, Skov helped Silkeborg avoid relegation by becoming their leading scorer, by scoring 10 league goals from his position on the wing.

[9] He made his league debut for the club on 10 February 2018 in a 5–1 home victory over Randers, playing all ninety minutes of the match.

[10] He scored his first league goal for the club a little over two weeks later, on 25 February 2018 in a 1–0 home victory over Odense BK.

[14][15] By March 2019, Skov was rumoured to Tottenham Hotspur and Arsenal after putting in outstanding performances in the Superliga.

[16] In July 2019, Skov signed a contract with TSG Hoffenheim for a transfer fee of €10 million.

[26] In March 2019 Skov was selected for the Denmark squad for the UEFA Euro 2020 qualifying tournament.